Hello!
The word "that" is the determiner indicating the connection between "his uncle's thoughts/opinions" and "working for the government is the key to stability". These two parts of the sentences are not complete when separated.
A conjunction would be a word connecting two different clauses like "but", "although", "however", "because". For example:
I am very tired "but" I will still stay up to watch the movie with you.
The word "but" joins the two clauses "I am very tired" and "I will still stay up to watch the movie with you" together. These clauses would also be complete sentences without the conjunction "but" joining them like this:
"I am very tired."
"I will still stay up to watch the movie with you."
I hope that helps!